void	nag_zungbr (Nag_OrderType order, Nag_VectType vect, Integer m, Integer n, Integer k, Complex a[], Integer pda, const Complex tau[], NagError *fail)

3

Description

nag_zungbr (f08ktc) is intended to be used after a call to nag_zgebrd (f08ksc), which reduces a complex rectangular matrix

A

to real bidiagonal form

B

by a unitary transformation:

A = Q B P^{H}

. nag_zgebrd (f08ksc) represents the matrices

Q

and

P^{H}

as products of elementary reflectors.

This function may be used to generate

Q

P^{H}

explicitly as square matrices, or in some cases just the leading columns of

Q

or the leading rows of

P^{H}

The various possibilities are specified by the arguments vect, m, n and k. The appropriate values to cover the most likely cases are as follows (assuming that

A

was an

m

n

matrix):

1.	To form the full $m$ by $m$ matrix $Q$ : nag_zungbr(order,Nag_FormQ,m,m,n,...) (note that the array a must have at least $m$ columns).
2.	If $m > n$ , to form the $n$ leading columns of $Q$ : nag_zungbr(order,Nag_FormQ,m,n,n,...)
3.	To form the full $n$ by $n$ matrix $P^{H}$ : nag_zungbr(order,Nag_FormP,n,n,m,...) (note that the array a must have at least $n$ rows).
4.	If $m < n$ , to form the $m$ leading rows of $P^{H}$ : nag_zungbr(order,Nag_FormP,m,n,m,...)

4

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5

Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $vect$ – Nag_VectTypeInput

On entry: indicates whether the unitary matrix

Q

P^{H}

is generated.

$vect = Nag_FormQ$: $Q$ is generated.
$vect = Nag_FormP$: $P^{H}$ is generated.

Constraint:

vect = Nag_FormQ

Nag_FormP

3: $m$ – IntegerInput

On entry:

m

, the number of rows of the unitary matrix

Q

P^{H}

to be returned.

Constraint:

m \geq 0

4: $n$ – IntegerInput

On entry:

n

, the number of columns of the unitary matrix

Q

P^{H}

to be returned.

Constraints:

$n \geq 0$ ;
if $vect = Nag_FormQ$ and $m > k$ , $m \geq n \geq k$ ;
if $vect = Nag_FormQ$ and $m \leq k$ , $m = n$ ;
if $vect = Nag_FormP$ and $n > k$ , $n \geq m \geq k$ ;
if $vect = Nag_FormP$ and $n \leq k$ , $n = m$ .

5: $k$ – IntegerInput

On entry: if

vect = Nag_FormQ

, the number of columns in the original matrix

A

vect = Nag_FormP

, the number of rows in the original matrix

A

Constraint:

k \geq 0

6: $a [\dim]$ – ComplexInput/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgebrd (f08ksc).

On exit: the unitary matrix

Q

P^{H}

, or the leading rows or columns thereof, as specified by vect, m and n.

order = Nag_ColMajor

, the

(i, j)

th element of the matrix is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

, the

(i, j)

th element of the matrix is stored in

a [(i - 1) \times pda + j - 1]

7: $pda$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array a.

Constraint:

pda \geq \max (1, m)

8: $tau [\dim]$ – const ComplexInput

Note: the dimension, dim, of the array tau must be at least

$\max (1, \min (m, k))$ when $vect = Nag_FormQ$ ;
$\max (1, \min (n, k))$ when $vect = Nag_FormP$ .

On entry: further details of the elementary reflectors, as returned by nag_zgebrd (f08ksc) in its argument tauq if

vect = Nag_FormQ

, or in its argument taup if

vect = Nag_FormP

9: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_ENUM_INT_3: On entry, $vect = 〈value〉$ , $m = 〈value〉$ , $n = 〈value〉$ and $k = 〈value〉$ .
Constraint: $n \geq 0$ and
if $vect = Nag_FormQ$ and $m > k$ , $m \geq n \geq k$ ;
if $vect = Nag_FormQ$ and $m \leq k$ , $m = n$ ;
if $vect = Nag_FormP$ and $n > k$ , $n \geq m \geq k$ ;
if $vect = Nag_FormP$ and $n \leq k$ , $n = m$ .
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 0$ .

On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pda \geq \max (1, m)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computed matrix

Q

differs from an exactly unitary matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision. A similar statement holds for the computed matrix

P^{H}

8

Parallelism and Performance

nag_zungbr (f08ktc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zungbr (f08ktc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The total number of real floating-point operations for the cases listed in Section 3 are approximately as follows:

1.	To form the whole of $Q$ : $\frac{16}{3} n (3 m^{2} - 3 m n + n^{2})$ if $m > n$ , $\frac{16}{3} m^{3}$ if $m \leq n$ ;
2.	To form the $n$ leading columns of $Q$ when $m > n$ : $\frac{8}{3} n^{2} (3 m - n)$ ;
3.	To form the whole of $P^{H}$ : $\frac{16}{3} n^{3}$ if $m \geq n$ , $\frac{16}{3} m^{3} (3 n^{2} - 3 m n + m^{2})$ if $m < n$ ;
4.	To form the $m$ leading rows of $P^{H}$ when $m < n$ : $\frac{8}{3} m^{2} (3 n - m)$ .

The real analogue of this function is nag_dorgbr (f08kfc).

10

Example

For this function two examples are presented, both of which involve computing the singular value decomposition of a matrix

A

, where

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i & - 0.05 + 0.41 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i & - 0.81 + 0.56 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i & - 1.11 + 0.60 i \\ - 0.37 + 0.38 i & 0.19 - 0.54 i & - 0.98 - 0.36 i & 0.22 - 0.20 i \\ 0.83 + 0.51 i & 0.20 + 0.01 i & - 0.17 - 0.46 i & 1.47 + 1.59 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i & 0.26 + 0.26 i \end{array})

in the first example and

A = (\begin{array}{r} 0.28 - 0.36 i & 0.50 - 0.86 i & - 0.77 - 0.48 i & 1.58 + 0.66 i \\ - 0.50 - 1.10 i & - 1.21 + 0.76 i & - 0.32 - 0.24 i & - 0.27 - 1.15 i \\ 0.36 - 0.51 i & - 0.07 + 1.33 i & - 0.75 + 0.47 i & - 0.08 + 1.01 i \end{array})

in the second.

A

must first be reduced to tridiagonal form by nag_zgebrd (f08ksc). The program then calls nag_zungbr (f08ktc) twice to form

Q

and

P^{H}

, and passes these matrices to nag_zbdsqr (f08msc), which computes the singular value decomposition of

A

NAG Library Function Document

nag_zungbr (f08ktc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results